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GEOMETRY Chapter 3: Angle Pairs, Lines and Planes

Parallel and Perpendicular Lines

Parallel lines are coplanar lines that do not intersect. Intersecting lines are coplanar and have exactly one point in commonIf intersecting lines meet at right angles, they are perpendicular, otherwise they are oblique.

Algebraic Property Two non-vertical lines are parallel if and only if they have the same slope.

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Parallel linesPerpendicular lines Obilque lines

Transitive Property of Parallel Lines

Let l1 , l2, l3 be three coplanar lines. Prove that if l1 is parallel to l2 and l2 is parallel to l3 then l1 is parallel to l3.

Solution: one way to write a proof is to superimpose A coordinate plane over the three lines. Do this in a way so that none of the lines is vertical as shown.

Given: l1 ║ l2 and l2 ║ l3 Prove: l1 ║ l3

Let m1, m2 and m3 be the slopes of l1 , l2, l3

Because l1 and l2 have the same slope, they are parallel.Theorem 3.1 Transitivity of Parallel LinesIf two lines are parallel to the same line, then they are parallel to each other

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m1 = m2 Parallel lines have the same slope

m2 = m3 Parallel lines have the same slope

m1 = m3 Transitivity Property of Equality

> >

l1 l2 l3

A Property of Perpendicular Lines

Let l1 , l2, l3 be coplanar. Prove that if l1 is perpendicular to l2 and l2 is perpendicular to l3 then l1 is parallel to l3.

Solution: one way is to superimpose a coordinate plane over the three lines. Do this in a way so that none of the lines is vertical or horizontal as shown.

Given: l1 l2 and l2 l3 Prove: l1 l3

Let m1, m2 and m3 be the slopes of l1 , l2, l3

Because l1 and l2 have the same slope, they are parallel.Theorem 3.2 Property of Perpendicular LinesIf two coplanar lines are perpendicular to the same line, then they are parallel to each other.

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m1• m2 = – 1 Product of slopes of lines is – 1

m2• m3 = – 1 Product of slopes of lines is – 1

m1• m2 = m2•m3 Transitivity Property of Equality

m1 = m3 Divide both sides by m2

l2 l1

l3

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Theorem 3.1 Transitivity of Parallel LinesIf 2 lines are parallel to the same line, then they are parallel to each other.

Algebraic Property2 non-vertical lines are parallel if and only if they have the same slope

Theorem 3.2 Property of Perpendicular LinesIf 2 coplanar lines are perpendicular to the same line, then they are parallel to each other

Algebraic Property2 non-vertical lines are parallel if and only if they have the same slope

Geometric DefinitionSkew lines are lines that do not lie in the same plane

Geometric DefinitionParallel planes are planes that do not intersect

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Proving that 2 lines are Parallel

Postulate 16 Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel

Alternate Interior Angles Converse if two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel

Consecutive Interior Angles Converse if two lines are cut by a transversal so that consecutive interior angles are supplementary , then the lines are parallel

Alternate Exterior Angles Converse if two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel

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Solving Systems of Linear Equations 3.2

In Algebra, 3 possibilities exist for a system of 2 linear equations: 1. One solution where the lines intersect2. No solution where lines are parallel3. Infinite solutions where the lines are the same line (called “coincident lines”)

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Postulate 12. If 2 distinct lines intersect, then their intersection is exactly one point.

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Solving Systems of Linear Equations 3.2

Solution: Begin by solving one of the equations for one of the variables. Then substitute into the other equation.

Example 1: Solve the system, x – 2 y = – 7 3 x + 4 y = 9

x = 2 y – 7 Solve the 1st equation for x

3 ( 2 y – 7 ) + 4 y = 9 Substitute into 2nd equation

6 y – 21 + 4 y = 9 Simplify

10 y = 30 Simplify

y = 3 Divide both sides by 10(– 1, 3 ) •

x – 2 y = 7

3x + 4y = 9

By substituting y = 3 into the first equation, you obtain x = – 1. Check this solution by substituting x = – 1 and y = 3 into both of the original equations. You can also check the solution graphically, as shown to the right.

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Finding a Parallel Line 3.2

Solution: Line l1 has a slope of – 3. Because parallel lines have the same slope, you know that l2 also has a slope of – 3 . You can find the y-intercept of l2 as follows:

Example 2: The line l1 is given by y = – 3x + 2. The line l2 is parallel to line l1 and passes through the point (2, 1). Find an equation for l2

y = m x + b Slope-intercept form

1 = (– 3 )(2) + b Substitute – 3 for m, 2 for x and 1 for y

1 = – 6 + b Simplify

7 = b Solve for b

y = – 3 x + 7 Equation for l2

(2, 1)•y = – 3 x + 2

Now that you know the slope and y-intercept of l2 you can write its equation.

Check that (2, 1) is on the line. The graph of l2 is shown to the right.

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l2 l1

Finding a Perpendicular Line 3.2

Solution: Line l1 has a slope of – 2. Because perpendicular lines have slopes that are negative reciprocals, you know that l2 also has a slope of ½ . You can find the y-intercept of l2 using the slope-intercept form as follows:

Example 3: The line l1 is given by y = – 2x + 1. The line l2 is perpendicular to line l1 and passes through the point (4, 0). Find an equation for l2

y = m x + b Slope-intercept form

0 = ( ½ )(4) + b Substitute ½ for m, 4 for x and 0 for y

0 = 2 + b Simplify

– 2 = b Solve for b

y = ½ x – 2 Equation for l2

(4, 0)

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y = – 2 x + 1

Now that you know the slope and y-intercept of l2 you can write its equation.

The graph of l2 is shown to the right.

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l2

l1

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H

GF

E

D

CB

A

AD and FG are parallelAD and AE are perpendicularAD and HG are skew

Relationships among lines 3.2

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Theorem: Through a given point on a line, there exists exactly one perpendicular to the given line, L

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Theorem: Through a given point NOT on a line, there exists exactly one perpendicular to the given line, L

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L

L

Perpendicular line

Perpendicular line

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Angles Formed by a Transversal

A transversal is a line that intersects two or more coplanar lines at different points.

Angles Formed by a TransversalIn the figure at the right, the transversal t intersects the lines l and m

Two angles are corresponding angles if they occupy corresponding positions, such as, < 1 and < 5

Two angles are alternate interior angles if they lie between l and m on opposite sides of t, such as < 2 and < 8

Two angles are alternate exterior angles if they lie outside l and m on opposite sides of t, such as < 1 and < 7

Two angles are consecutive interior angles if they lie between l and m on same side of t, such as < 2 and < 5

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t

l

m 6

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7

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Angles Formed by a Transversal

Postulate 15 Corresponding Angles Postulate. If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Interior Angles Theorem if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent

Consecutive Interior Angles Theorem if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary

Alternate Exterior Angles Theorem if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

Perpendicular Transversal Theorem if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the second

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Given: < 1 < 2

PROVE: L 1 ║ L 2

Statements Reasons

< 1 < 2 Given

< 1 < 3 Vertical angles are congruent

< 3 < 2 Transitive Property of Congruence

L 1 ║ L 2 Corresponding < ‘ s are lines are ║

Proof of “Alternate Interior Angles Converse”

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1

t

3 L 1

L 2

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VECTORS

A Vector is a quantity that has both a direction and magnitude (length).

Vectors are represented by directed line segments from an initial point to a terminal point.

2 Vectors are parallel if the have the same or opposite directions

NOTE: a ray and a vector both have an INITIAL point but a vector also has a TERMINAL point whereas a ray does not.

2 Vectors are equal if they have the same length and direction, however they don’t have to have the same initial and terminal points.

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rayvector

initial

terminal

initial

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VECTORS

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Q

P

(x2 , y2)

(x1 , y1)

Let P (x1 , y1) and Q (x2 , y2) be points in a coordinate plane.

The Vector PQ can be represented by the directed line segment whose initial point is ( 0, 0 ) and whose terminal point is ( x2 ─ x1 , y2 ─ y1 ). This ordered pair representation is

denoted by v = ( x2 ─ x1 , y2 ─ y1 ).

( x2 ─ x1 , y2 ─ y1 )

(0 , 0)

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Writing a vector as an ordered pair

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Q

P(4 , 2)

(3 , 4)

Let P (3 , 4) and Q (4 , 2) Find the ordered pair representation of PQ where the initial point is ( 0, 0 ) and the terminal point is ( 1, ─ 2 )

The Vector PQ = ( x2 ─ x1 , y2 ─ y1 ) = ( 4 ─3 , 2 ─ 4 ) = ( 1 , ─ 2 )

denoted by v = ( x2 ─ x1 , y2 ─ y1 ).

( 1, ─ 2 )

(0 , 0)

Let P (3 , 4) and Q (4 , 2) Use Distance Formula:

PQ = ( 4 ─ 3 ) 2 + ( 2 ─ 4 ) 2

PQ = 55

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ADDING VECTORS

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•v

u

(0 , 0)

Two vectors can be added to form a new vector using the parallelogram rule:To geometrically add u and v, move the initial point of u to the terminal point of v or vice versa. Note that the sum of u + v is the DIAGONAL of a parallelogram.

v

u

v + u

Algebraically: the sum of v = ( a1 , b1 ) and u = ( a2 , b2 ) is

v + u = (a1 + a2 , b1 + b2 )

If v + u = ( 0 , 0 ), the v and u have the same length but opposite directions

Example above:the sum of v = ( 1 , 5 ) and u = ( 4 , 2 ) is

v + u = ( 1 + 4 , 5 + 2 ) = ( 5 , 7 )

If v + u = ( 0 , 0 ), the v and u have the same length but opposite directions

(4 , 2)

(1 , 5)

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MULTIPLYING VECTORS

Two vectors can be multiplied using the operation of dot product. The result is a real number, NOT a vector.

Let v = ( a1 , b1 ) and u = ( a2 , b2 ). The DOT PRODUCT of these 2 vectors is

v ● u = a1 b1 + a2 b2

Two non-zero vectors are perpendicular if and only if their dot product is zero.

v = ( 5 , 10 ) and u = ( 4 , ─ 2 )

v ● u = (5) (4) + (10) (─ 2) = 20 + ─ 20 = 0

v and u are

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(4 , ─ 2 )

( 5 , 10 )

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To find: SUM of 2 VECTORS: v = (1,5) and u = (4,2)

ADD the x coordinates and y coordinates to get SUM coordinatesv + u = (a1 + a2 , b1 + b2 )

v + u = ( 1 + 4 , 5 + 2 ) = (5 ,7)

To find: Ordered Pair Representation given an initial point (1,5 ) and terminal point (4,2)

SUBTRACT the initial coordinates from the terminal coordinates(4 – 1 , 2 – 5)

(3 , – 3)

To find: DOT PRODUCT of 2 VECTORS: v = (1,5) and u = (4,2)

MULTIPLY the x coordinates and, then multiply the y coordinates and then ADD both to get the Product

v • u = (a1 • a2 , b1 • b2 )v • u = ( 1 • 4 + 5 • 2 ) = 14

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Distance: The term distance in geometry is always interpreted as the shortest path between two points

A B

Distance: The shortest distance between two points is a straight line.

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